This is a proposal to bring techniques from group theory and (noncommutative) harmonic analysis to bear on computational aspects of structural biology. Novel algorithms will be developed and implemented in order to address the following problems: de novo determination of protein structure from unassigned residual dipolar couplings without a prior knowledge of the Saupe alignment tensor; rapid minimization of the "rotation function" for molecular replacement of multi-domain proteins; de novo determination of electron densities from projections in cryo-electron-microscopy. Our unified approach casts these as minimization problems and fast functional evaluations on finite and Lie groups, for which we will need to generalize methods such as gradient descent and FFTs to the group-valued setting. Our team combines expertise in mathematics, engineering, and biology necessary to make progress in this highly interdisciplinary subject. The problem of protein structure determination is central to the understanding of protein function and molecular design. This is absolutely critical to the progress of the health and environmental sciences in regards to efforts related to "designer drugs" - i.e., the development of targeted therapies, be they for humans, animals, or plants. Our efforts have the potential to remake high-throughput techniques by providing experimentalists with new and efficient techniques for the comparison of molecular structures. The novelty of our approach is in its focus on the use of the tools of group theory and group representation theory (harmonic analysis) in this life sciences setting. A particularly attractive aspect of this proposal is the close connection between theory and practice. Our interdisciplinary team combines mathematical, computational, biological, and engineering skills - so that it is well-poised to make progress on a problem that is inherently multidisciplinary and one that draws on each of these areas